GMAT Math : Geometry

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Example Questions

Example Question #244 : Data Sufficiency Questions

You are given two rectangles, $\textup{Rect } ABCD$ and $\textup{Rect } EFGH$

True or false: $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

Statement 1: $\frac{AC}{EG} = \frac{BD}{FH}$

Statement 2: $\frac{AB}{EF} = \frac{BC}{FG}$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, AC = BD , EG = FH , and, as a consequence, $\frac{AC}{EG} = \frac{BD}{FH}$ regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.

Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.

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Example Question #245 : Data Sufficiency Questions

You are given two rectangles, $\textup{Rect } ABCD$ and $\textup{Rect } EFGH$ .

Let the perimeter of $\textup{Rect } ABCD$ be , and let the perimeter of $\textup{Rect } EFGH$ be .

Let the area of $\textup{Rect } ABCD$ be and the area $\textup{Rect } EFGH$ be .

True or false: $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

Statement 1: $\left (\frac{p }{q} \right ) ^{2}= \frac{r}{s}$

Statement 2: $\frac{AC}{EG} = \frac{p}{q}$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true.

Examine these three rectangles. The one of the left is $\textup{Rect } ABCD$ ; the other two have the same dimensions, and both are called $\textup{Rect } EFGH$ , except that the names of the vertices are differently arranged:

Rectangles

Regardless of which $\textup{Rect } EFGH$ is chosen:

The ratio of the perimeters is $\frac{14}{28} = \frac{1}{2}$ ;

The ratio of the areas is $\frac{12}{48} = \frac{1}{4} =\left ( \frac{1 }{2} \right )^{2}$ ; and,

The ratio $\frac{AC}{EG} = \frac{5}{10} = \frac{1}{2}$ .

The conditions of the problem are met for both pairings, but in one case, $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ and in the other, $\textup{Rect } ABCD \nsim \textup{Rect } EFGH$ (that is, the rectangles are similar but the similarity statement given may be true or false).

The two statements together provide insufficient information.

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Example Question #1 : Dsq: Calculating The Perimeter Of A Polygon

Given a regular hexagon and a regular octagon, which, if either, has the greater perimeter?

Statement 1: The sidelength of the octagon is one foot.

Statement 2: The sidelength of the hexagon is fifteen inches.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Each of the two statements allows you to find out the perimeter of one of the polygons by multiplying its sidelength by the number of its sides. However, neither statement offers any clues to the perimeter of the other polygon. Both statements together, however, allow you to determine and to compare both perimeters.

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Example Question #1 : Polygons

You are given a regular hexagon and a regular pentagon. Which one has the greater perimeter?

Statement 1: The hexagon and the pentagon have the same area

Statement 2: The apothem of the hexagon is greater than that of the pentagon

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The area of a polygon is one-half the product of its perimeter and its apothem (the perpendicular distance from the center to a side). Therefore,

$A = \frac{1}{2}ap$ ,

$\frac{1}{2}ap =A$

$\frac{1}{2}ap \cdot \frac{2}{a} =A\cdot \frac{2}{a}$

$p = \frac{2A}{a}$

Therefore, the perimeter can be determined from both area and apothem, but not from one alone. Neither statement alone gives you enough information, but from both statements together, it can be determined that the pentagon has the greater perimeter.

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Example Question #2 : Polygons

Untitled

Note: Figure NOT drawn to scale. All angles shown are right angles.

What is the perimeter of the above figure?

Statement 1: a = 20

Statement 2: b = 40

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The figure can be seen as a $d \times e$ rectangle cut out of an $a \times b$ rectangle.

Untitled

The perimeter of the composite figure is

P = a + b + c + d + e + f .

However, since opposite sides of a rectangle are congruent, then, as can be seen in the figure, c + e = b and d+f = a .

The perimeter can then be rewritten:

P = a + b + c + d + e + f

= a + d + f + b + c + e

= a + a + b + b

=2a + 2b

Therefore, it is necessary and sufficient to know and ; the other four sidelengths are not needed to determine the perimeter of the figure.

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Example Question #3 : Dsq: Calculating The Perimeter Of A Polygon

Consider regular decagon JKLMNOPQRS .

I) Side is 56 inches long.

II) Side plus Side is equivalent to 112 inches.

Find the perimeter of JKLMNOPQRS .

Possible Answers:

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Correct answer:

Either statement is sufficient to answer the question.

Explanation:

To find perimeter, we need the length of all the sides. Note that we are dealing with a regular polygon, so all of its sides are equal.

Statement I gives us one side length. Multiply the side by ten (decagons have ten sides):

$p=56\cdot 10=560$

Statement II gives us the length of two sides together. Multiply by 5 to get the total perimeter:

$p=112\cdot 5=560$

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Example Question #5 : Dsq: Calculating The Perimeter Of A Polygon

Consider pentagon SPLIT .

I) Side has the same length as side , 5 inches.

II) Side is one-third the length of side , and sides and have the same length as side .

What is the perimeter of SPLIT ?

Possible Answers:

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Correct answer:

Both statements are needed to answer the question.

Explanation:

Consider pentagon SPLIT .

I) Side has the same length as side , 5 inches.

II) Side is one-third the length of sides , and .

What is the perimeter of SPLIT ?

To find perimeter, we need to know the length of all the sides.

Statement I gives us the lengths of two sides.

Statement II relates one of the sides given in Statement I to the other three sides:

$3TS=PL=LI=LT=3\cdot 5=15$

P=5+5+15+15+15=55in

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Example Question #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

Hexagon_44

The hexagon in the above diagram is regular. If $\overline{AB}$ has length 10, which of the following expressions is equal to the length of $\overline{AC}$ ?

Possible Answers:

$5\sqrt{6}$

$10\sqrt{6}$

$10\sqrt{3}$

$10\sqrt{2}$

Correct answer:

$10\sqrt{3}$

Explanation:

The answer can be seen more easily by constructing the altitude of $\bigtriangleup ABC$ from , as seen below:

Untitled

Each interior angle of a hexagon measures $120^{\circ }$ ,and the altitude also bisects $\angle ABC$ , the vertex angle of isosceles $\bigtriangleup ABC$ . $\bigtriangleup ABM$ is easily proved to be a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem,

$BM = \frac{1}{2} \cdot AB = \frac{1}{2} \cdot 10 = 5$

$AM = BM \cdot \sqrt{3} = 5\sqrt{3}$

The altitude also bisects $\overline{AC}$ at , so

$AC = 2 \cdot AM = 2 \cdot 5 \sqrt{3} = 10 \sqrt{3}$ .

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Example Question #252 : Data Sufficiency Questions

Calculate the diagonal of a rectangle.

Statement 1: The perimeter is 10.

Statement 2: The area is 4.

Possible Answers:

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

$\textup{EACH statement ALONE is sufficient.}$

Correct answer:

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

Explanation:

Statement 1: The perimeter is 10.

Given the perimeter of a rectangle is 10, set up the equation such that the sum of twice of length and width are equal to 10.

2L+2W=10

There are multiple combinations of length and width that will work in this scenario.

Statement 2: The area is 4.

Write the formula for area of a rectangle and substitute the area.

A=LW

4=LW

Statements 1 and 2 require each other in order to solve for the length and width since there are two unknown variables.

Afterward, the Pythagorean Theorem can be used to solve for the diagonal of the rectangle.

Therefore:

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

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Example Question #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

A regular pentagon has been drawn on the side of a building by some mathematically minded graffiti artists. What is the length of a diagonal across it?

1) The length of a side is 5feet .

2) Each of the internal angles is 108 degrees.

Possible Answers:

Either of the statements is sufficient.

Statement 2 alone is sufficient.

Neither of the statements, separate or together, is sufficient.

Statement 1 alone is sufficient.

Together the two statements are sufficient.

Correct answer:

Statement 1 alone is sufficient.

Explanation:

If the pentagon is regular, then it is known that each of the internal angles is 108 degrees, since the sum of the five interior angles of a pentagon is 540 degrees. Statement 2 does not give new information, nor does it give enough.

Statement 1 does, however. The length of the diagonal can be found by using the law of cosines:

c^2=a^2+b^2-2abcos(C)

a=5, b=5, C=108

and c is the length of the diagonal.

$d=\sqrt{5^2+5^2-2(5)(5)cos(108)}$

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #244 : Data Sufficiency Questions

Example Question #245 : Data Sufficiency Questions

Example Question #1 : Dsq: Calculating The Perimeter Of A Polygon

Example Question #1 : Polygons

Example Question #2 : Polygons

Example Question #3 : Dsq: Calculating The Perimeter Of A Polygon

Example Question #5 : Dsq: Calculating The Perimeter Of A Polygon

Example Question #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

Example Question #252 : Data Sufficiency Questions

Example Question #1 : Dsq: Calculating The Length Of A Diagonal Of A Polygon

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